For many years, Hilbert held the position at the Mathematical Institute at the University of Göttingen that was recognized as the most prestigious mathematical position in Germany, and possibly, in the world.
His mathematical accomplishments, methods, and influence have been the subject of much study and writing, all well represented by two particular works:
Constance Reid, Hilbert, Springer-Verlag, New York-Heidelberg-Berlin, 1970;
Hermann Weyl, "David Hilbert and His Mathematical Work," Bulletin of the American Mathematical Society 60 (1944) 612-654.
A shortened version of Weyl's article is reprinted in Reid's book, which is also the source of the image of Hilbert seen here. The image dates from 1912, a time when portraits of professors were popular on postcards sold in Göttingen.
"How can it be allowed that a woman become a Privatdozent? Having become a Privatdozent, she can then become a professor and a member of the University Senate. Is it permitted that a woman enter the Senate?" They argued informally, "What will our soldiers think when they return to the University and find that they are expected to learn at the feet of a woman?""Meine Herren," countered Hilbert, "I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bathhouse." Noether's application was rejected, but Hilbert arranged for her to stay at Göttingen by having her lectures announced under his name.
Hans Freudenthal writes in Biographical Dictionary of Mathematics (Charles Scribner's Sons, New York, 1991):
Hilbert's scientific activity can be roughly divided into six periods, according to the years of publication of the results: up to 1893 (at Königsberg), algebraic forms; 1894-1899, algebraic number theory; 1899-1903, foundations of geometry; 1904-1909, analysis (Dirichlet's principle, calculus of variations, integral equations, Waring's problem); 1912-1914, theoretical physics; after 1918, foundations of mathematics.These are similar to Weyl's placement of five main periods of Hilbert's publications: "i. Theory of invariants (1885-1893). ii. Theory of algebraic number fields (1893-1898). iii. Foundations, (a) of geometry (1898-1902), (b) of mathematics in general (1922-1930). iv. Integral equations (1902-1912). v. Physics (1910-1922). The headings are a little more specific than they ought to be," Weyl continues, "Not all of Hilbert's algebraic achievements are directly related to invariants. His papers on calculus of variations are here lumped together with those on integral equations. And of course there are some overlappings and a few stray children who break the rules of time, the most astonishing his proof of Waring's theorem in 1909."
Hilbert's collected works have been repeatedly published; for example,
David Hilbert, Gesammelte Abhandlungen, 3 volumes, Chelsea, New York, 1965 (vol. 1, Zahlentheorie, 539 pages; vol. 2, Algebra, 453 pages; vol. 3, Analysis, 435 pages).
His books and books co-authored with others have also been reprinted in various languages, several during the 1990's. These may be viewed using FirstSearch or some other database.
Among other works about David Hilbert, his times, and his influence, are
Constance Reid, Courant in Göttingen and New York, The Story of an Improbably Mathematician, Springer-Verlag, New York-Heidelberg-Berlin, 1976;
D. E. Rowe, "Klein, Hilbert, and the Göttingen mathematical tradition," Osiris 5 (1989) 186-213;
Leo Corry, "David Hilbert and the Axiomatization of Physics," Archives for History of Exact Sciences 51 (1997) 83-198.